Absolute continuity of self-similar measures, their projections and convolutions

Abstract

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand's projection theorem for planar self-similar sets.

33 pages, no figures

References in corpus (1)

• On the dimensions of a family of overlapping self-affine carpets

Cited by in corpus (13)

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• Absolute continuity of non-homogeneous self-similar measures
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• Absolute continuity of complex Bernoulli convolutions
• On the projections of mutual multifractal spectra
• On the projections of the multifractal Hewitt-Stromberg dimension functions
• On the interior of projections of planar self-similar sets
• Sums of two homogeneous Cantor sets
• Lyapunov exponents for products of matrices
• Singular non-Pisot Bernoulli convolutions
• On multifractal formalism for self-similar measures with overlaps
• Dimension conservation for self-similar sets and fractal percolation